Theorem B..7 (Error Bound for Lipschitz Condition)
If
is a continuous function satisfying the
Lipschitz condition

on , then the inequality

holds.

Proof.
Abbreviating notation we set
. We will use the Lipschitz condition,
Corollary A.7, and
Lemma B.4.

This completes the proof. Q.E.D.

Theorem B..8 (Asymptotic Formula)
Let
be a function and , then

Proof.
We define the vector through
, where the
are the integers over which we sum in
. Using
Theorem A.14 we see

where
. Summing this equation like the sum
in
we obtain

since many terms vanish or can be summed because of
Lemma B.4. Noting
we can
apply the same technique as in the proof of
Theorem B.5 for estimating the last sum
in the last equation, i.e., splitting the sum into two parts for
and
. Hence we see that for all
this sum is less equal
for all sufficiently
large , which yields the claim. Q.E.D.